Sufficient Optimality and Sensitivity Analysis of a Parameterized Min-Max Programming

Abstract

Sufficient optimality and sensitivity of a parameterized min-max programming with fixed feasible set are analyzed. Based on Clarke's subdifferential and Chaney's second-order directional derivative, sufficient optimality of the parameterized min-max programming is discussed first. Moreover, under a convex assumption on the objective function, a subdifferential computation formula of the marginal function is obtained. The assumptions are satisfied naturally for some application problems. Moreover, the formulae based on these assumptions are concise and convenient for algorithmic purpose to solve the applications.

1. Introduction

In this paper, sufficient optimality and sensitivity analysis of a parameterized min-max programming are given. The paper is triggered by a local reduction algorithmic strategy for solving following nonsmooth semi-infinite min-max-min programming (SIM³P, see [1, 2], etc. for related applications reference):ms.ti.n_𝑥𝑔𝑓(𝑥(𝑥))=max_𝑦∈𝑌min_{1≤𝑖≤𝑞}𝑔_𝑖(𝑥,𝑦)≤0.(1.1) With the local reduction technique, the SIM³P can be rewritten as a bilevel programming first, where the lower problem is the following parameterized min-max programming 𝑃_𝑥 (see [3–5] for related reference of local reduction strategy): sm.ti.n_𝑦𝑦𝑔∈(𝑥𝑌,𝑦.)=max_{1≤𝑖≤𝑞}−𝑔_𝑖(𝑥,𝑦)(1.2) To make the bilevel strategy applicable to SIM³P, it is essential to discuss the second-order sufficient optimality of 𝑃_𝑥 and give sensitivity analysis of the parameterized minimum 𝑦(𝑥) and corresponding marginal function 𝑔(𝑥,𝑦(𝑥)).

Sensitivity analysis of optimization problems is an important aspect in the field of operation and optimization research. Based on different assumptions, many results on kinds of parametric programming have been obtained ([6–9], etc.). Among these, some conclusions on parameterized min-max programming like (1.2) have also been given. For example, based on variation analysis, parameterized continuous programming with fixed constraint was discussed in [7]. Problem like (1.2) can be seen as a special case. Under the inf-compactness condition and the condition objective function is concave with respect to the parameter, directional derivative computational formula of marginal function for (1.2) can be obtained directly. However, concave condition cannot be satisfied for many problems. Recently, Fréchet subgradients computation formula of marginal functions for nondifferentiable programming in Asplund spaces was given ([9]). By using Fréchet subgradients computation formula in [9], subgradient formula of marginal function for (1.2) is direct. But the formula is tedious, if utilizing the formula to construct optimality system of (1.1), the system is so complex that it is difficult to solve the obtained optimality system.

For more convenient computational purpose, the focus of this paper is to establish sufficient optimality and simple computation formula of marginal function for (1.2). Based on Clarke's subdifferential and Chaney's second-order directional derivative, sufficient optimality of the parameterized programming 𝑃_𝑥 is given first. And then Lipschitzian continuousness of the parameterized isolated minimizer 𝑦(𝑥) and the marginal function 𝑔(𝑦(𝑥),𝑥) is discussed; moreover, subdifferential computation formula of the marginal function is obtained.

2. Main Results

Let 𝑌 in (1.2) be defined as 𝑌={𝑦∈𝑅^𝑚∶ℎ_𝑖(𝑦)≤0,𝑖=1,…,𝑙}, where ℎ_𝑖(⋅) and 𝑖=1,…,𝑙, are twice continuously differentiable functions on 𝑅^𝑚, and 𝑔_𝑖(⋅,⋅) in (1.2) are twice continuously differentiable functions on 𝑅^𝑛×𝑚. In the following, we first give the sufficient optimality condition of (1.2) based on Clarke's subdifferential and Chaney's second-order directional derivative, and then make sensitivity analysis of the parameterized problem 𝑃_𝑥.

2.1. Sufficient Optimality Conditions of 𝑃_𝑥

Definition 2.1 (see [10]). For a given parameter 𝑥, a point 𝑦^∗∈𝑌 is said to be an local minimum of problem 𝑃_𝑥 if there exists a neighborhood 𝑈 of 𝑦^∗ such that 𝑔(𝑥,𝑦)≥𝑔𝑥,𝑦^∗,∀𝑦∈𝑈∩𝑌,𝑦≠𝑦^∗.(2.1)

Assumption 2.2. For a given parameter 𝑥, suppose that 𝑃_𝑥 satisfying the following constraint qualification: 𝑑∈𝑅^𝑚∶∇ℎ_𝑖(𝑦)^𝑇𝑑<0,∀𝑖∈𝐼_ℎ(𝑦),𝑦∈𝑌≠∅,(2.2) where 𝐼_ℎ(𝑦)={𝑖={1,…,𝑙}∶ℎ_𝑖(𝑦)=0}.
For a given parameter 𝑥, denote the Lagrange function of 𝑃_𝑥 as 𝐿(𝑥,𝑦,𝜆)=𝑔(𝑥,𝑦)+∑^𝑙𝑖=1𝜆_𝑖ℎ_𝑖(𝑦), then the following holds.

Theorem 2.3. For a given parameter 𝑥, if 𝑦^∗ is a minimum of 𝑃_𝑥, Assumption 2.2 holds, then there exists a 𝜆^∗∈𝑅^𝑙+ such that 0∈𝜕_𝑦𝐿(𝑥,𝑦^∗,𝜆), where 𝜕_𝑦𝐿(𝑥,𝑦^∗,𝜆^∗) denotes the Clarke's subdifferential of 𝐿(𝑥,𝑦^∗,𝜆^∗). Specifically, the following system holds: 0∈𝜕_𝑦𝑔𝑥,𝑦^∗+^𝑙_𝑖=1𝜆_𝑖∇ℎ_𝑖𝑦^∗,(2.3) where 𝜕_𝑦𝑔(𝑥,𝑦) denotes Clarke's subdifferential of 𝑔(𝑥,𝑦) with respect to 𝑦, it can be computed as co{∇_𝑦𝑔_𝑖(𝑥,𝑦^∗)∶𝑖∈𝐼(𝑥,𝑦^∗)}, co{⋅} is an operation of making convex hull of the elements, 𝐼(𝑥,𝑦^∗)={𝑖∈{1,…,𝑞}∶𝑔(𝑥,𝑦^∗)=𝑔(𝑥,𝑦^∗)}.

Proof. The conclusion is direct from Theorem 3.2.6 and Corollary 5.1.8 in [11].

Since 𝑔(𝑥,𝑦)=max_{1≤𝑖≤𝑝}{𝑔_𝑖(𝑥,𝑦)} is a directional differentiable function (Theorem 3.2.13 in [11]), the directional derivative of 𝑔(𝑥,𝑦) with respect to 𝑦 in direction 𝑑 can be computed as follows: 𝑔^𝑦(𝑥,𝑦;𝑑)=max𝜉^𝑇𝑑∶∀𝜉∈𝜕_𝑦𝑔(𝑥,𝑦),∀𝑑∈𝑅^𝑚.(2.4)

Definition 2.4 (see [10]). Let 𝑓(𝑥) is a locally Lipschitzian function on 𝑅^𝑛, 𝑢 be a nonzero vector in 𝑅^𝑛. Suppose that 𝑑∈𝜕_𝑢𝑓(𝑥)=𝜐∈𝑅^𝑛∶∃𝑥_𝑘,𝜐_𝑘,s.t.𝑥⟶_𝑘𝑥,𝜐_𝑘⟶𝜐,𝜐_𝑘∈𝜕𝑓𝑥_𝑘foreach𝑘(2.5) define Chaney's lower second-order directional derivative as follows: 𝑓^−(𝑥,𝜐,𝑢)=liminf𝑓𝑥^𝑘−𝑓(𝑥)−𝜐^𝑇𝑥_𝑘−𝑥𝑡^2𝑘,(2.6) taking over all triples of sequences {𝑥_𝑘}, {𝜐_𝑘}, and {𝑡_𝑘} for which (a)𝑡_𝑘>0 for each 𝑘 and {𝑥_𝑘}→𝑥; (b)𝑡_𝑘→0 and (𝑥_𝑘−𝑥^∗)/𝑡_𝑘 converges to 𝑢; (c){𝜐_𝑘}→𝜐 with 𝜐_𝑘∈𝜕𝑓(𝑥_𝑘) for each 𝑘.
Similarly, Chaney's upper second-order directional derivative can be defined as 𝑓^+=limsup𝑓𝑥^𝑘−𝑓(𝑥)−𝜐^𝑇𝑥_𝑘−𝑥𝑡^2𝑘,(2.7) taking over all triples of sequences {𝑥_𝑘}, {𝜐_𝑘}, and {𝑡_𝑘} for which (a), (b), and (c) above hold.
For parameterized max-type function 𝑔(𝑥,𝑦)=max_{1≤𝑖≤𝑝}{−𝑔_𝑖(𝑥,𝑦)}, where 𝑥 is a given parameter, its Chaney's lower and upper second-order directional derivatives can be computed as follows.

Proposition 2.5 (see [12]). For any given parameter 𝑥, Chaney's lower and upper second-order directional derivatives of 𝑔(𝑥,𝑦) with respect to 𝑦 exist; moreover, for any given 0≠𝑢∈𝑅^𝑞, 𝜐∈𝜕_𝑢𝑔(𝑥,𝑦), it has 𝑔^−(𝑥,𝑦;𝑑)=min12^𝑞_𝑖=1𝑔𝑎_𝑗𝑢^𝑇∇^2𝑦𝑔_𝑖(𝑥,𝑦)𝑢∶𝑎∈𝑇_𝑢(𝑔,𝑦,𝜐)_+,(𝑥,𝑦;𝑑)=max12^𝑞_𝑖=1𝑎_𝑗𝑢^𝑇∇^2𝑦𝑔_𝑖(𝑥,𝑦)𝑢∶𝑎∈𝑇_𝑢(𝑔,𝑦,𝜐),(2.8) where𝑇_𝑢(𝑔,𝑦,𝜐)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∃𝑦^(𝑘)(1)𝑦_(𝑘),𝑎^(𝑘),𝜐^(𝑘),suchthat𝑎∈𝑅_𝑞+⟶𝑦indirection𝑢,∶((32))𝑎𝜐^(𝑘)_(𝑘)⟶𝜐,and𝜐^(𝑘)∈𝜕_𝑦𝑔𝑥,𝑦^(𝑘),𝑘=1,2,…,(4)𝑎_𝑗(𝑘)⟶𝑎,𝑎^(𝑘)∈𝐸_𝑞,𝜐^(𝑘)=∑^𝑝_𝑖=1𝑎^𝑖(𝑘)∇_𝑦𝑔_𝑖𝑥,𝑦^(𝑘),=0,for𝑗∉𝐾_𝑔𝑦^(𝑘)⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭,(2.9) where 𝐾_𝑔(𝑦^(𝑘))={𝑖∈𝑄∶𝑔_𝑖(𝑥,𝑦^(𝑛))=𝑔(𝑥,𝑦^(𝑛)),∃𝑦^(𝑛)∈𝐵(𝑦,1/𝑛),∀𝑛∈𝑁}, 𝐸_𝑞={𝑎∈𝑅^𝑞+∶∑^𝑝𝑖=1𝑎_𝑖=1}, 𝑄={1,…,𝑞}, and 𝐵(𝑦,1/𝑛) denotes the ball centered in 𝑦 with radius 1/𝑛.

Theorem 2.6 (sufficiency theorem). For a given parameter 𝑥∈𝑅^𝑛, Assumption 2.2 holds, then there exists 𝑦^∗∈𝑅^𝑚 such that (2.3) holds. Moreover, for any feasible direction 𝑑∈𝑅^𝑚 of 𝑌, that is, max{∇ℎ_𝑖(𝑦^∗)^𝑇𝑑∶1≤𝑖≤𝑙}≤0, if 𝑑 satisfying one of the following conditions: (1)𝑔^𝑦(𝑥,𝑦^∗;𝑑)≠0; (2)𝑔^𝑦(𝑥,𝑦^∗;𝑑)=0, ∑^𝑙𝑖=1𝜆_𝑖∇ℎ_𝑖(𝑦)^𝑇𝑑=0, that is, 𝐿^𝑦(𝑥,𝑦;𝑑)=0, and min12^𝑞_𝑖=1𝑎_𝑖𝑑^𝑇∇^2𝑦𝑔_𝑖𝑥,𝑦^∗𝑑∶𝑎∈𝐸_𝑞+^𝑙_𝑖=1𝜆_𝑖𝑑^𝑇∇²ℎ_𝑖𝑦^∗𝑑>0,(2.10)then 𝑦^∗ is a local minimum of 𝑃_𝑥.

Proof. (1) If not, then there exists sequences 𝑡_𝑘↓0, 𝑑_𝑘→𝑑, 𝑦_𝑘=𝑦^∗+𝑡_𝑘𝑑_𝑘∈𝑌 such that 𝑔𝑥,𝑦_𝑘<𝑔𝑥,𝑦^∗.(2.11) As a result, 𝑔^𝑦(𝑥,𝑦^∗;𝑑)=lim_𝑡↓0(𝑔(𝑥,𝑦^∗+𝑡𝑑)−𝑔(𝑥,𝑦^∗))/𝑡=lim_𝑘→+∞(𝑔(𝑥,𝑦^∗+𝑡_𝑘𝑑_𝑘)−𝑔(𝑥,𝑦^∗))/𝑡_𝑘≤0. If 𝑔^𝑦(𝑥,𝑦^∗;𝑑)≠0, then 𝑔^𝑦(𝑥,𝑦^∗;𝑑)<0. From (2.4), we know that 𝜉^𝑇𝑑<0forall𝜉∈𝜕_𝑦𝑔(𝑥,𝑦^∗). Hence, for the direction 𝑑∈𝑅^𝑚, we have 𝜉^𝑇𝑑+^𝑙_𝑖=1∇ℎ_𝑖𝑦^∗^𝑇𝑑<0,𝜉∈𝜕_𝑦𝑔𝑥,𝑦^∗.(2.12) On the other hand, from 𝑦^∗ satisfying (2.3), we know that there exists a 𝜉∈𝜕_𝑦𝑔(𝑥,𝑦^∗) such that 𝜉^𝑇𝑑+^𝑙_𝑖=1∇ℎ_𝑖𝑦^∗^𝑇𝑑=0,(2.13) which leads to a contradiction to (2.12).
(2) From Theorem 4 in [10] and Proposition 2.5, the conclusion is direct.

2.2. Sensitivity Analysis of Parameterized 𝑃_𝑥

In the following, we make sensitivity analysis of parameterized min-max programming 𝑃_𝑥, that is, study variation of isolated local minimizers and corresponding marginal function under small perturbation of 𝑥.

For convenience of discussion, for any given parameter 𝑥, denote 𝑦^∗(𝑥) as a minimizer of 𝑃_𝑥, 𝜐(𝑥)=min{𝑔(𝑥,𝑦)∶𝑦∈𝑌} as the corresponding marginal function value and make the following assumptions first.

Assumption 2.7. For given 𝑥∈𝑅^𝑛, the parametric problem 𝑃_𝑥 is a convex problem, specifically, 𝑔_𝑖(𝑥,𝑦) and 𝑖=1,…,𝑞 are concave functions with respect to that variables 𝑦 and ℎ_𝑗(𝑦),𝑗=1,…,𝑙 are convex functions.

Assumption 2.8. Let 𝐼_ℎ(𝑦)={𝑖∈𝐿∶ℎ_𝑖(𝑦)=0}, {∇ℎ_𝑖(𝑦)∶𝑖∈𝐼_ℎ(𝑦)} are linearly independent.

Definition 2.9 (see Definition 2.1, [13]). For a given 𝑥, 𝑦∈𝑌 is said to be an isolated local minimum with order 𝑖 (𝑖 = 1 or 2) of 𝑃_𝑥 if there exists a real 𝑚>0 and a neighborhood 𝑉 of 𝑦 such that 𝑔𝑥,𝑦>𝑔𝑥,𝑦+12𝑚‖‖𝑦−𝑦‖‖^𝑖,∀𝑦∈𝑉∩𝑌,𝑦≠𝑦.(2.14)

Theorem 2.10. For a given 𝑥∈𝑅^𝑛, Assumptions 2.2–2.8 hold, then the following conclusions hold: (1)if 𝑦^∗(𝑥) with corresponding multiplier 𝜆^∗ is the solution of (2.3), then 𝑦^∗(𝑥) is a unique first-order isolated minimizer of 𝑃_𝑥;(2)for any minimum 𝑦^∗(𝑥), it is a locally Lipschitzian function with respect to 𝑥, that is, there exists a 𝐿₁>0, 𝛿>0 such that ‖‖𝑦^∗𝑥^𝑘−𝑦^∗(𝑥)‖‖≤𝐿₁‖‖𝑥^𝑘−𝑥‖‖,∀𝑥^𝑘∈𝑈(𝑥,𝛿),𝑦^∗𝑥^𝑘∈𝑌𝑥^𝑘,(2.15) where 𝑌(𝑥^𝑘) denotes minima set of 𝑃_𝑥𝑘;(3)for any minimum 𝑦^∗(𝑥), marginal function 𝜐(𝑥)=𝑔(𝑥,𝑦^∗(𝑥)) is also a locally Lipschitz function with respect to 𝑥, and 𝜕𝜐(𝑥)⊆𝑆(𝑥), where 𝑆(𝑥)=co∇_𝑥𝑔_𝑖𝑥,𝑦^∗(𝑥),𝑖∈𝐼𝑥,𝑦^∗(𝑥),(2.16) and 𝐼(𝑥,𝑦^∗(𝑥))={𝑖∈{1,…,𝑞}∶𝑔_𝑖(𝑥,𝑦^∗(𝑥))=𝑔(𝑥,𝑦^∗(𝑥))}. As a result, 𝜕𝜐(𝑥)=_{𝑖∈𝐼(𝑥,𝑦∗(𝑥))}𝜆_𝑖∇_𝑥𝑔_𝑖𝑥,𝑦^∗(𝑥)∶𝜆_𝑖≥0,_{𝑖∈𝐼(𝑥,𝑦∗(𝑥))}𝜆_𝑖=1.(2.17)

Proof. (1) From Assumption 2.7, it is direct that 𝑦^∗(𝑥) is a global minimizer of 𝑃_𝑥. We only prove 𝑦^∗(𝑥) is a first-order isolated minimizer.
If the conclusion does not hold, then there exists a sequence {𝑦^𝑘}∈𝑌(𝑥) converging to 𝑦^∗(𝑥), 𝑦^𝑘≠𝑦^∗(𝑥), and a sequence 𝑚_𝑘, 𝑚_𝑘>0, and 𝑚_𝑘 converges to 0 such that 𝑔𝑥,𝑦^𝑘≤𝑔𝑥,𝑦^∗(𝑥)+12𝑚^𝑘‖‖𝑦^𝑘−𝑦^∗‖‖,𝑦^𝑘∈𝑌.(2.18) Take 𝑑_𝑘=(𝑦^𝑘−𝑦^∗(𝑥))/‖𝑦^𝑘−𝑦^∗(𝑥)‖, for simplicity, we suppose 𝑑_𝑘→𝑑, with ‖𝑑‖=1. Let 𝑡_𝑘=‖𝑦^𝑘−𝑦^∗(𝑥)‖, then from 𝑦^𝑘∈𝑌, 𝑑_𝑘→𝑑 and 𝑌 is compact, we have 𝑦^∗(𝑥)+𝑡_𝑘𝑑∈𝑌,𝑡_𝑘⟶0,(2.19) that is, ∇ℎ_𝑖𝑦^∗(𝑥)^𝑇𝑑≤0,∀𝑖∈𝐼𝑥,𝑦^∗(𝑥).(2.20) From Assumption 2.8, we know that ∑_{𝑖∈𝐼(𝑥,𝑦∗(𝑥))}∇ℎ_𝑖(𝑦^∗(𝑥))^𝑇𝑑≠0. As a result, we have ∑_{𝑖∈𝐼(𝑥,𝑦∗(𝑥))}∇ℎ_𝑖(𝑦^∗(𝑥))^𝑇𝑑<0.
From the first equation of (2.3), we know that there exists a 𝑧∈𝜕_𝑦𝑔(𝑥,𝑦^∗(𝑥)) such that for any feasible direction 𝑑, 𝑧^𝑇𝑑=−∑_{𝑖∈𝐼(𝑥,𝑦∗(𝑥))}𝜆_𝑖∇ℎ_𝑖(𝑦^∗(𝑥))^𝑇𝑑>0. Hence, 𝑔^𝑦𝑥,𝑦^∗(𝑥);𝑑=max𝜉^𝑇𝑑∶𝜉∈𝜕_𝑦𝑔𝑥,𝑦^∗(𝑥)≥𝑧^𝑇𝑑>0.(2.21) On the other hand, from 𝑦^∗(𝑥) is a minimizer, we know that 𝑔^𝑦(𝑥,𝑦^∗(𝑥);𝑑)≥0, this leads to a contradiction;
(2) from Assumption 2.8 and Theorem 3.1 in [13], the conclusion is direct;
(3) since 𝑔(𝑥,𝑦) is a locally Lipschitzian function with respect to 𝑥 and 𝑦, then there exists 𝛿>0, 𝛿′>0, and 𝐿₂>0 such that for any 𝑥₁∈𝑈(𝑥,𝛿), 𝑦∈𝑈(𝑦^∗(𝑥),𝛿^), it has ||𝑔||𝑔𝑥(𝑥₁,,𝑦𝑦^∗)(−𝑥)𝑔−𝑥,𝑔𝑦_∗𝑥,𝑦^∗(𝑥)||≤𝐿₂‖‖𝑥₁−𝑥‖‖,(𝑥)||≤𝐿₂‖𝑦−𝑦^∗(𝑥)‖.(2.22) As to 𝑥₁∈𝑈(𝑥,𝛿), from the conclusion in (1.2), there exists a a 𝐿₁>0 such that ‖𝑦^∗(𝑥₁)−𝑦^∗(𝑥)‖≤𝐿₁‖𝑥₁−𝑥‖. As a result, ||𝜐𝑥=₁||𝑔−𝜐𝑥(𝑥)₁||=||𝑔𝑥,𝑦^∗𝑥₁−𝑔𝑥,𝑦^∗(𝑥)₁||≤||𝑔𝑥,𝑦^∗𝑥₁−𝑔𝑥₁,𝑦^∗(𝑥)+𝑔𝑥₁,𝑦^∗(𝑥)−𝑔𝑥,𝑦^∗(𝑥)₁||≤𝐿₂,𝑦^∗𝑥₁−𝑔𝑥₁,𝑦^∗(𝑥)||+||𝑔𝑥₁,𝑦^∗(𝑥)−𝑔𝑥,𝑦^∗(𝑥)||‖‖𝑦^∗𝑥₁−𝑦^∗(𝑥)‖‖+𝐿₂‖‖𝑥₁−𝑥‖‖≤𝐿₂1+𝐿₁‖‖𝑥₁−𝑥‖‖.(2.23) Hence, the marginal function 𝜐(𝑥) is a local Lipschitzian function with respect to 𝑥.
Let 𝑆(𝑥)={−∇_𝑥𝑔_𝑖(𝑥,𝑦(𝑥)),𝑖∈𝐼(𝑥,𝑦(𝑥))}, then 𝑆(𝑥)=co{𝜉,𝜉∈𝑆(𝑥)}. We prove that 𝑆(𝑥) is closed first, that is, prove for any sequence {𝑥^𝑘}⊂𝑅^𝑛, 𝑥^𝑘→𝑥, 𝑧^𝑘∈𝑆(𝑥^𝑘), 𝑧^𝑘→𝑧, it has 𝑧∈𝑆(𝑥).
From 𝑧^𝑘∈𝑆(𝑥^𝑘), there exist 𝑦^𝑘∈𝑌(𝑥^𝑘); 𝑖_𝑘∈𝐼(𝑥^𝑘,𝑦^𝑘) such that 𝑧^𝑘=−∇_𝑥𝑔_𝑖𝑘(𝑥^𝑘,𝑦^𝑘). Without loss of generality, suppose that {𝑦^𝑘} converges to 𝑦; {𝑖_𝑘} converges to 𝑖. From Proposition 3.3 in [14], it has 𝑦∈𝑌(𝑥), and 𝑖∈𝐼(𝑥,𝑦) and from ∇_𝑥𝑔_𝑖(𝑥,𝑦) is a continuous function, it has 𝑧=lim_𝑘→+∞𝑧_𝑘=lim_𝑘→+∞∇_𝑥𝑔_𝑖𝑘(𝑥^𝑘,𝑦^𝑘)=∇_𝑥𝑔_𝑖(𝑥,𝑦)∈𝑆(𝑥). As a result, 𝑆(𝑥) is a closed set.
From Theorem 3.2.16 in [11], for any 𝜉∈𝜕𝜐(𝑥), there exists 𝑥^𝑘∈𝑅^𝑛, 𝑥^𝑘→𝑥 such that ∇𝜐(𝑥^𝑘) exists and 𝜉=lim_𝑘→+∞∇𝜐(𝑥^𝑘). In addition, for arbitrary 𝑑∈𝑅^𝑛, it has ∇𝜐𝑥^𝑘^𝑇𝑑=𝜐^𝑥^𝑘;𝑑=lim_𝑡↓0𝜐𝑥^𝑘+𝑡𝑑−𝜐𝑥^𝑘=lim_𝑡↓0𝑡𝑔𝑥^𝑘+𝑡𝑑,𝑦^∗𝑥^𝑘+𝑡𝑑−𝑔𝑥^𝑘,𝑦^∗𝑥^𝑘≤lim_𝑡↓0𝑡𝑔𝑥^𝑘+𝑡𝑑,𝑦^∗𝑥^𝑘−𝑔𝑥^𝑘,𝑦^∗𝑥^𝑘=max_{𝑥𝑖∈𝐼𝑘,𝑦}𝑡−∇^𝑥𝑔_𝑖𝑥^𝑘,𝑦^𝑇𝑑.(2.24) From the definition of 𝑆(𝑥^𝑘), ∃𝑧^𝑘∈𝑆(𝑥^𝑘) such that 𝑧^𝑘𝑇𝑑=max_{𝑖∈𝐼(𝑥𝑘,𝑦)}{−∇_𝑥𝑔_𝑖(𝑥^𝑘,𝑦)^𝑇𝑑}. Hence, it has ∇𝜐(𝑥^𝑘)^𝑇𝑑≤𝑧^𝑘𝑇𝑑.
From 𝑧^𝑘→𝑧∈𝑆(𝑥)⊂𝑆(𝑥), ∇𝜐(𝑥^𝑘)→𝜉 and ∇𝜐(𝑥^𝑘)^𝑇𝑑≤𝑧^𝑘𝑇𝑑, it has 𝜉^𝑇𝑑≤𝑧^𝑇𝑑, that is, for arbitrary 𝑑∈𝑅^𝑛 and 𝜉∈𝜕𝜐(𝑥), there exists 𝑧∈𝑆(𝑥) such that 𝜉^𝑇𝑑≤𝑧^𝑇𝑑.
If 𝜕𝜐(𝑥)⊂𝑆(𝑥) does not hold, then there exists a 𝜉∈𝜕𝜐(𝑥) and 𝜉∉𝑆(𝑥). From 𝑆(𝑥) is a compact convex set and separation theorem ([15]), there exists a 𝑑∈𝑅^𝑛 such that 𝜉^𝑇𝑑<0 and for arbitrary 𝑧∈𝑆(𝑥), 𝑧^𝑇𝑑≥0, which leads to a contradiction. As a result, 𝜕𝜐(𝑥)⊂𝑆(𝑥) holds. From 𝜕𝜐(𝑥)⊂𝑆(𝑥) and 𝑆(𝑥)=co{∇_𝑥𝑔_𝑖(𝑥,𝑦^∗(𝑥)),𝑖∈𝐼(𝑥,𝑦^∗(𝑥))}, computation formula (2.17) is direct.

3. Discussion

In this paper, sufficient optimality and sensitivity analysis of a parameterized min-max programming are given. A rule for computation the subdifferential of 𝜐(𝑥) is established. Though the assumptions in this paper are some restrictive compared to some existing work, the assumptions hold naturally for some applications. Moreover, the obtained computation formula is simple, it is beneficial for establishing a concise first-order necessary optimality system of (1.1), and then constructing effective algorithms to solve the applications.

Acknowledgments

This research was supported by the National Natural Science Foundation of China no. 11001092 and the Fundamental Research Funds for the Central Universities no. 2011QC064.